First-principles calculations of the spin-orbit coupling in graphene with hydrogen adatoms in dense and dilute limits are presented. The chemisorbed hydrogen induces a giant local enhancement of spin-orbit coupling due to sp(3) hybridization which depends strongly on the local lattice distortion. Guided by the reduced symmetry and the local structure of the induced dipole moments, we use group theory to propose realistic minimal Hamiltonians that reproduce the relevant spin-orbit effects for both single-side semi-hydrogenated graphene (graphone) and for a single hydrogen adatom in a large supercell. The principal linear spin-orbit band splittings are driven by the breaking of the local pseudospin inversion symmetry and the emergence of spin flips on the same sublattice.